Math 251 Final Exam Review Fall 2016

Transcription

1 Below are a set of review problems that are, in general, at least as hard as the problems you will see on the final eam. You should know the formula for area of a circle, square, and triangle. All other area formulae needed on the eam will be given to you. 1. Find the indicated limit. e 2 1 (a) lim (b) lim ln(sin()) π t 2 4 (c) lim t 2 t v (d) lim v v (e) lim 3 (f) lim (g) lim (h) lim 1 (i) lim e 2 (j) ( lim (1 + sin())cot() Let g(t) = t 2. (a) For what values of t is g(t) continuous? (b) For what values of t is g(t) differentiable? ) 1

2 3. Consider the graph of y = f(), shown below on the domain [ 5, 5]. [Recall that represents a point that is defined on the graph, while represents a point that is not defined on the graph.] y (a) Compute each of the following limits, if possible. If a limit does not eist, write DNE. i. lim 2 f() ii. lim f() 0 iii. lim f() 1 iv. lim f() 1 (b) Is f continuous at = 1? Use limits to justify your conclusion. (c) For which value(s) of is f not differentiable? 4. Find the indicated derivative. (a) f (), where f() = ( ) 3 (b) g (t), where g(t) = t + t 1 ln(t) (c) y, where y = e sin(3θ) (d) h (θ), where h(θ) = cos 2 (tan(θ)) (e) f (), where f() = e 2 ln(tan()) (f) dy d, where y4 + 2 y = + 3y (g) d dt, where t2 cos() + sin(3t) = t 2

3 5. Find an equation of the tangent line to the curve at the given value of. (a) y = (2 + )e ; = 0 (b) y = ; = 0 (c) 3 y 5y = 4; = 1 6. At what value(s) of is the tangent line to the curve f() = 4+3e (1+ 2 ) horizontal? 7. For each function below: i. Find all intercepts. ii. Find the vertical and horizontal asymptotes, if any. iii. Find the intervals of increase or decrease. iv. Find the local maimum and minimum values. v. Find the intervals of concavity and the inflection points. Then use this information to sketch a graph of the function. (a) f() = (b) f() = Find all local etrema of the function. Then find the absolute etrema of the function on the given interval. [You should include both where the etrema occur, as well as the values of the etrema.] (a) f() = 1 on [ 1, 1] (b) f() = ln() on [1, 3] (c) f() = 3 4 on [ 2, 2] Dan, the owner of the Dan s Pizza Pi company, is reviewing his daily sales figures and pricing structure for his specialty pizza. He has found that in order to sell pizzas he needs to charge p() dollars for each pizza where p() = 40e 0.01 for values of in the interval [0, 150]. (a) Find the function R() that gives Dan s revenue from the sales of his specialty pizza. [Hint: Revenue is price multiplied by quantity sold.] (b) It is in Dan s best interest to maimize the amount of revenue he makes each day. How many pizzas must Dan sell to maimize revenue? (c) What price does Dan need to charge for each pizza in order to sell the number of pizzas that will maimize his revenue? 10. Adam and Justin leave a cafe at 2pm. Adam walks directly west at a rate of 3 miles per hour. Justin walks directly south at a rate of 4 miles per hour. 3

4 (a) How fast is the distance between Adam and Justin changing at 3pm? (b) Gavin is standing 0.75 miles east of the cafe making a phone call. From where Gavin is standing, he can measure the angle between the street and Justin, as Justin walks south. How fast is this angle changing at 2:15pm? (c) At 2pm, Deb is 5 miles north of the cafe and begins to ride her bike. She rides directly east at a pace of 9 miles per hour. How is the distance between Adam and Deb changing at 3pm? 11. The volume of a right circular cone is 1 3 πr2 h, where r is the radius of the base and h is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) Find the rate of change of the volume with respect to the radius if the height is constant. (c) Find the rate of change of the volume with respect to time, when the radius is 3 m and the height is 4 m. Assume the rate of change happens equally for r and h with a magnitude of 2 m/s. 12. A conical water tank with verte down has a radius of 4 m at the top and is 16 m high. Assume water is being pumped into the tank at a rate of 2 m 3 /min. (a) Find the rate at which the water level is rising when the water is 3 m deep. (b) Find the rate at which the water level is rising after 2 3 π minutes. (c) What is the depth of the water at the moment when the water level is rising at a rate of 4 m/min? 13. A particle is moving along the curve y = + 1. As the particle passes through the point (3, 2), its -coordinate increases at a rate of 2 cm/s. How fast is the distance from the particle to the origin changing at this instant? 14. A demographic study of a certain city indicates that P (r) hundred people live r miles from the civic center, where 5(3r + 1) P (r) = r 2 + r + 2 (a) For what values of r is P (r) increasing? For what values is it decreasing? (b) At what distance from the civic center is the population largest? What is this largest population? 15. A 5-year projection of population trends suggests that t years from now, the population of a certain community will be P (t) = t 3 + 9t t + 50 thousand. At what time during the 5-year period will the population be growing most rapidly? 4

5 16. Find two positive real numbers whose product is 7 and whose sum is as small as possible. 17. A cylindrical can (with a top and bottom) is to be made with 600 π cm 2 of aluminum. Find the dimensions (in cm) that will maimize the volume of the can. [The volume of a cylinder is πr 2 h and the surface area of the sides of a cylinder equals 2πrh.] 18. Find the shortest distance from the point (5, 0) to the curve 2y 2 = If 4800 cm 2 of material is available to make a bo with a square base and an open top, find the largest possible volume of the bo. 20. A manufacturing company is asked to create special bricks for a new retaining wall to be built in Eugene. The bricks are to be rectangular with the length equal to three times the width. The top and bottom of the bricks is to be coated with a reinforced material that costs 10 cents per in 2. The other sides of the bricks will be coated with a standard coating that costs 6 cents per in 2. Find the dimensions of the bricks that will minimize the cost of production if the bricks must have volume 50 in Use the function f() = at a = 0 to estimate the value of linear approimation formula by using a 22. Use the function f() = 1 to estimate the value of 4.4 using a linear approimation formula. 23. Give an approimation for ln(0.9) using a linear approimation formula. 5